Abstract
Syrnrnet:n'c spaces is a special topic in Riemannian geometry .
These spaces werefirst studied and classified by Elie Cartan . _
In this research we study in a logical ordering their snucture
through manifolds , Lie groups , Lie algebras and basics of Riemannian
geometry. _
The study covers locally and globally symmetric spaces , endowed
with some examples for them such as Euclidean spaces , spheres
hyperbolic spaces and some applications on locally syrrunetric spaces in
the field of arithmetic and algebraic groups , including quadratic and
modular fonns , lattices , the realization of discrete series representations
of groups , Poincare and linear symmetric spaces .
Their classification is also discussed by introducing compact and
noncompact symmetric spaces besides the types ( l , ll , Ill , IV ) .
This classification is carried through Lie algebras , root systems and their
Dynkin diagrams .
The aim of the research is to display in a simplified manner the
connection between symmetric spaces and the differential geometric
temis such as manifolds , Lie groups , Lie algebras and basics of
Riemannian geometry with some of the applications of symmetric spaces.
